Theorem genpprecl 5087

Description: Pre-closure law for general operation on positive reals.

Hypothesis

Ref	Expression
genp.1	|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}

Assertion

Ref	Expression
genpprecl	|- ((A e. P. /\ B e. P.) -> ((C e. A /\ D e. B) -> (CGD) e. (AFB)))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,w,v,u,G,y,z

Proof of Theorem genpprecl

Step	Hyp	Ref	Expression
1		eqid 1474	. 2 |- (CGD) = (CGD)
2		genp.1	. . . . . 6 |- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}
3	2	genpv 5085	. . . . 5 |- ((A e. P. /\ B e. P.) -> (AFB) = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))})
4	3	eleq2d 1539	. . . 4 |- ((A e. P. /\ B e. P.) -> ((CGD) e. (AFB) <-> (CGD) e. {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))}))
5		oprex 3978	. . . . 5 |- (CGD) e. V
6		eqeq1 1479	. . . . . . 7 |- (f = (CGD) -> (f = (gGh) <-> (CGD) = (gGh)))
7	6	anbi2d 615	. . . . . 6 |- (f = (CGD) -> (((g e. A /\ h e. B) /\ f = (gGh)) <-> ((g e. A /\ h e. B) /\ (CGD) = (gGh))))
8	7	2exbidv 1280	. . . . 5 |- (f = (CGD) -> (E.gE.h((g e. A /\ h e. B) /\ f = (gGh)) <-> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh))))
9	5, 8	elab 1894	. . . 4 |- ((CGD) e. {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))} <-> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh)))
10	4, 9	syl6bb 535	. . 3 |- ((A e. P. /\ B e. P.) -> ((CGD) e. (AFB) <-> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh))))
11		eleq1 1532	. . . . . . 7 |- (g = C -> (g e. A <-> C e. A))
12		eleq1 1532	. . . . . . 7 |- (h = D -> (h e. B <-> D e. B))
13	11, 12	bi2anan9 631	. . . . . 6 |- ((g = C /\ h = D) -> ((g e. A /\ h e. B) <-> (C e. A /\ D e. B)))
14		opreq12 3965	. . . . . . 7 |- ((g = C /\ h = D) -> (gGh) = (CGD))
15	14	eqeq2d 1484	. . . . . 6 |- ((g = C /\ h = D) -> ((CGD) = (gGh) <-> (CGD) = (CGD)))
16	13, 15	anbi12d 627	. . . . 5 |- ((g = C /\ h = D) -> (((g e. A /\ h e. B) /\ (CGD) = (gGh)) <-> ((C e. A /\ D e. B) /\ (CGD) = (CGD))))
17	16	cla42egv 1861	. . . 4 |- ((C e. A /\ D e. B) -> (((C e. A /\ D e. B) /\ (CGD) = (CGD)) -> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh))))
18	17	anabsi5 495	. . 3 |- (((C e. A /\ D e. B) /\ (CGD) = (CGD)) -> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh)))
19	10, 18	syl5bir 210	. 2 |- ((A e. P. /\ B e. P.) -> (((C e. A /\ D e. B) /\ (CGD) = (CGD)) -> (CGD) e. (AFB)))
20	1, 19	mpan2i 698	1 |- ((A e. P. /\ B e. P.) -> ((C e. A /\ D e. B) -> (CGD) e. (AFB)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  {cab 1462  E.wrex 1644  (class class class)co 3958  {copab2 3959  P.cnp 4968

This theorem is referenced by:  genpnmax 5093  addclprlem2 5102  mulclprlem 5104  distrlem1pr 5110  distrlem2pr 5111  ltaddpr 5123  ltexprlem7 5131

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fv 3194  df-opr 3960  df-oprab 3961

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